Computational Algebraic Geometry
Forward from the Editor This issue of the Journal of Symbolic Computation presents selected papers on work presented at, or related to, the meeting Michigan Computational Algebraic Geometry (MCAG 2012), held in Rochester, Michigan, May 26-28, 2012. MCAG conferences are a yearly event held in one of the public universities in Michigan. Their aim is to bring together algebraic geometers from academia and industry and to foster further cooperation. MCAG-group is a network of algebraic geometers in Michigan and surrounding areas in order to encourage collaboration, exploration of new applications, and stronger connections to industry. The group organizes yearly conferences in different Michigan institutions, training sessions for students, and encourages collaboration among different mathematics departments from Michigan. During the first MCAG conference, held at Oakland University, there were over 100 participants from all public research universities in Michigan. There were also participants from Ontario, Canada and many Midwest sates. Among notable speakers were I. Dolgachev (University of Michigan), W. Fulton (University of Michigan), G. Freudenburg (Western Michigan University), R. Kulkarni (Michigan State University), H. Derksen (University of Michigan), C. Wampler (General Motors), K. Lee (Wayne State University), L. Li (Oakland University), Z. Teiltler (Boise State University), et al. The second MCAG conference will be held in Kalamazoo (Western Michigan University) during the month of May, 2013. While the focus of MCAG conferences is on computational aspects of algebraic geometry, we encourage participants and talks from all areas of algebraic geometry. There are of particular interest the directions of the classical algebraic geometry such as invariant theory, geometry of algebraic curves, automorphisms of curves and surfaces, moduli spaces, Jacobians of curves, Abelian varieties, theta functions and their applications to different areas such as integrable systems, physics, coding theory and cryptography,
Preprint submitted to Journal Symbolic Computation
April 21, 2013
and others. The articles selected for this special issue represent this wide variety of interests. Here is a summary of the topics covered in this issue. • Herbert Lange and Peter Newstead study the Clifford index for vector bundles on smooth projective curves of genus g ≥ 4. Two definitions of the Clifford index for vector bundles on a smooth projective curve C of genus g ≥ 4 were introduced in a previous paper by the authors. In another paper the authors obtained results on one of these indices for bundles of rank 3. Here we extend these results in the case where C has classical Clifford index 3. In particular they prove Mercat’s conjecture for bundles of rank 3 for g ≤ 8 and g ≥ 13 when C has classical Clifford index 3. Complete results are obtained in the case of genus 7. • Gene Freudenburg’s paper lays out the basic theory of the down operator D of the polynomial ring R = k[x0 , x1 , x2 , . . .], defined by Dxi = xi−1 (i ≥ 1) and Dx0 = 0, where k is any field of characteristic zero. The only linear invariant is x0 , and the quadratic invariants are well known and easily described. One of the paper’s main results gives a complete description of the cubic invariants, ordered according to bidegree and the number of variables involved. The distinction between core and compound invariants is introduced, and quartic and quintic invariants are studied relative to this property. As an application of the theory, a new family of counterexamples to Hilbert’s Fourteenth Problem is given; the proof of non-finite generation is much simpler than for previously known examples. • Cynthia Vinzant and Daniel Plaumann study the determinantal representations of hyperbolic plane curves. In 2007, Helton and Vinnikov proved that every hyperbolic plane curve has a definite real symmetric determinantal representation. By allowing for Hermitian matrices instead, the authors are able to give a new proof that relies only on the basic intersection theory of plane curves. They show that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extend a classical construction of determinantal representations of Dixon from 1902. Like the Helton-Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality. 2
• Milagros Izquierdo, Gabriel Bartolini, and Antonio Costa study automorphism groups of algebraic curves (compact Riemann surfaces) which have a degree p cyclic cover to P1 . In this work they obtain the group of conformal and anticonformal automorphisms of real cyclic pgonal Riemann surfaces, where p ≥ 3 is a prime integer and the genus of the surfaces is at least (p − 1)2 + 1. They use Fuchsian and NEC groups, and cohomology of finite groups. • Amod Agashe and Randy Heaton study computing intersection numbers between abelian varieties associated to subspaces of modular forms. They derive a formula for the intersection number between abelian varieties associated to complementary subspaces of the space of cuspidal modular forms. This formula can be used for computations using modular symbols. An implementation of this method is also discussed. My thanks go to the Office of the Provost and the Office of Research Administration at Oakland University for providing a grant for MCAG 2012, the Meadow Brook Hall Foundation for allowing us to use their wonderful facilities, the National Security Agency (NSA) for the conference grant we received in organizing this conference, and the Department of Mathematics and Statistics at Oakland University for handling all the administrative tasks of the conference. Special thanks go to the organizers of the MCAG 2012 for an excellent job in organizing the conference. Particularly, I want to thank Charles Wampler (General Motors), Dan Erman (University of Michigan), and Dan Steffy, Lubjana Beshaj, Fred Thompson (Oakland University) for their work and efforts in putting together MCAG 2012. We hope the reader enjoys this collection of papers and finds the MCAG conferences an interesting venue of cooperation.
T. Shaska Oakland University E-mail address:
[email protected]
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